No, the logic is that the US government may target enemies in war by all the means lawful against enemies inwar, even if those enemies are US citizens, anywhere in the world, including within the US.
And there's plenty of precedent for that.
There may be good reasons given the nature and mechanisms of modern warfare to establish some additional limits on that now, but we can't have the discussion effectively if we start with a false premise about the rationale and the false idea that prosecuting war without special limitations when some of the individual enemy combatants are citizens is novel.
> I wonder if POTUS has to at least sign all the kill orders.
In general, no, the President is not required to personally sign orders to kill enemies in war, regardless of their citizenship.